These are just some of the fragmented musings that I note down to help me structure my thinking, reasoning, and writing, and give me some vague sense of mental clarity. They are not meant to be complete, based on established knowledge, or coherent (to anyone but me). But if you find them somehow helpful…that’s great. Sharing these thoughts here with the hope of one day integrating them into a more complete manual for anyone who, like me, struggles to organize their brain.
(linguistic/mathematical/etc. expressions of ideas)
first-order statements
= basic true/false statements (e.g., x is/is not y)
note: can think of a statement like this as expressing that an object possesses a property or ‘does something’, which is roughly equivalent to an equivalence between object and property or object and operation), sometimes with categorical qualifiers (e.g., all, some, etc.)
note also: can think of these as definitions
more complex definitions (e.g., of more complex objects/operations) may combine multiple of these using logical connectives (see (3))
we can call these ‘compound statements’ of first-order statements
equivalences/implications
= combined first-order statements that are either equivalent to or follow from one another
note: can think of these as ‘theorems’ — where the equivalence (equality/inequality)/implication requires proof
note also: even first-order statements sometimes require proof (e.g., that an object/operation, or a particular subset or combination of objects/operations, has a particular property) = constructive proof
higher-order statements
= combinations of (1) and (2) using logical connectives (and, or — more accurately, conjunctions, since ‘logical connectives’ includes ‘not’ [‘truth qualifier’])
note: first-order statements can also contain lists — e.g., of objects, attributes, or details
just be careful to not list a series of distinct ideas that act more as clauses than list items and end up with an extremely cluttered sentence
All of the above can be viewed as elaborations on basic object (modifier)-verb (modifier) structure fundamental to most (if not all) modern human language using the explanatory and reasoning tools of grammar and logic.
(applies best to argumentative pieces/mathematics/other complex ideas)
(i.e., logical) structure (for expressing complex higher-order ideas)
(mathematical explanations)
topics/points/subpoints can be arranged
symmetry = restatement of an idea
points of a topic paragraph are symmetrical to the topic sentences of all support paragraphs